A new representation of Lagrange’s theorem in differential calculus
It is found a new representation of the mean value Lagrange’s theorem in the differential calculus. Any function increment can be expressed through the derivatives in the ending points of a given closed interval. Mean values of the Lagrange derivative and our theory derivative are coincided, but t...
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Інститут проблем штучного інтелекту МОН України та НАН України
2014
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irk-123456789-853042015-07-25T03:01:38Z A new representation of Lagrange’s theorem in differential calculus Mironenko, L.P. Petrenko, I.V. Обучающие и экспертные системы It is found a new representation of the mean value Lagrange’s theorem in the differential calculus. Any function increment can be expressed through the derivatives in the ending points of a given closed interval. Mean values of the Lagrange derivative and our theory derivative are coincided, but the middle points are different. Our theory allows easily find the middle point and it is not so easy according to Lagrange’s theorem. Furthermore, our theory makes it possible to formulate the second mean value theorem in integral calculus, as it is a consequence of differential theorem. У статті сформульовано нове представлення відомої теореми у диференційному численні про середнє – теореми Лагранжа. Прирощення функції представлено через похідні у кінцевих точках відрізку. Середнє значення похідної по Лагранжу і нашої теорії співпадають, але не співпадають середні точки. Наша теорія дозволяє знайти середню точку, що важко зробити на підставі теореми Лагранжа. Крім того, наша теорема дає можливість сформулювати теорему про середнє у інтегральному численні, бо вона просто є наслідком диференціальної теореми. В статье сформулировано новое представление известной теоремы дифференциального исчисления о среднем – теоремы Лагранжа. Приращение функции выражено через производные в концевых точках отрезка. По величине среднее значение производной по Лагранжу и по нашей теории совпадают, однако не совладают средние точки. Наша теория позволяет легко определить среднюю точку, что затруднительно в теореме Лагранжа. Кроме того, наша теория дает возможность сформулировать вторую теорему о среднем в интегральном исчислении, так как она является следствием дифференциальной теоремы. 2014 Article A new representation of Lagrange’s theorem in differential calculus / L.P. Mironenko, I.V. Petrenko // Искусственный интеллект. — 2014. — № 2. — С. 129–133. — Бібліогр.: 5 назв. — англ. 1561-5359 http://dspace.nbuv.gov.ua/handle/123456789/85304 514.116 en Искусственный интеллект Інститут проблем штучного інтелекту МОН України та НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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Обучающие и экспертные системы Обучающие и экспертные системы |
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Обучающие и экспертные системы Обучающие и экспертные системы Mironenko, L.P. Petrenko, I.V. A new representation of Lagrange’s theorem in differential calculus Искусственный интеллект |
| description |
It is found a new representation of the mean value Lagrange’s theorem in the differential calculus. Any
function increment can be expressed through the derivatives in the ending points of a given closed interval.
Mean values of the Lagrange derivative and our theory derivative are coincided, but the middle points are
different. Our theory allows easily find the middle point and it is not so easy according to Lagrange’s
theorem. Furthermore, our theory makes it possible to formulate the second mean value theorem in integral
calculus, as it is a consequence of differential theorem. |
| format |
Article |
| author |
Mironenko, L.P. Petrenko, I.V. |
| author_facet |
Mironenko, L.P. Petrenko, I.V. |
| author_sort |
Mironenko, L.P. |
| title |
A new representation of Lagrange’s theorem in differential calculus |
| title_short |
A new representation of Lagrange’s theorem in differential calculus |
| title_full |
A new representation of Lagrange’s theorem in differential calculus |
| title_fullStr |
A new representation of Lagrange’s theorem in differential calculus |
| title_full_unstemmed |
A new representation of Lagrange’s theorem in differential calculus |
| title_sort |
new representation of lagrange’s theorem in differential calculus |
| publisher |
Інститут проблем штучного інтелекту МОН України та НАН України |
| publishDate |
2014 |
| topic_facet |
Обучающие и экспертные системы |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/85304 |
| citation_txt |
A new representation of Lagrange’s theorem in differential calculus / L.P. Mironenko, I.V. Petrenko // Искусственный интеллект. — 2014. — № 2. — С. 129–133. — Бібліогр.: 5 назв. — англ. |
| series |
Искусственный интеллект |
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| first_indexed |
2025-07-06T12:30:55Z |
| last_indexed |
2025-07-06T12:30:55Z |
| _version_ |
1836900738518745088 |
| fulltext |
ISSN 1561-5359 «Штучний інтелект» 2014 № 2 129
5M
UDK 514.116
L.P. Mironenko, I.V. Petrenko
Donetsk National Technical University, Ukraine
Ukraine, 83000, Donetsk, Аrtema st., 58, mironenko.leon@yandex.ru
A New Representation of Lagrange’s Theorem
in Differential Calculus
Л.П. Мироненко, И.В. Петренко
Донецкий национальный технический университет, Украина
Украина, 83000, г. Донецк, ул. Артема, 58, mironenko.leon@yandex.ru
Новое представление теоремы Лагранжа
в дифференциальном исчислении
Л.П. Мироненко, І.В. Петренко
Донецький національний технічний університет, Україна
Україна, 83000, м. Донецьк, вул. Артема, 58
Нове представлення теореми Лагранжа
у диференційному численні
It is found a new representation of the mean value Lagrange’s theorem in the differential calculus. Any
function increment can be expressed through the derivatives in the ending points of a given closed interval.
Mean values of the Lagrange derivative and our theory derivative are coincided, but the middle points are
different. Our theory allows easily find the middle point and it is not so easy according to Lagrange’s
theorem. Furthermore, our theory makes it possible to formulate the second mean value theorem in integral
calculus, as it is a consequence of differential theorem.
Keywords: technique, theorem, mean value, function, integral, derivative, Lagrange’s theorem.
В статье сформулировано новое представление известной теоремы дифференциального исчисления о
среднем – теоремы Лагранжа. Приращение функции выражено через производные в концевых точках
отрезка. По величине среднее значение производной по Лагранжу и по нашей теории совпадают,
однако не совладают средние точки. Наша теория позволяет легко определить среднюю точку, что
затруднительно в теореме Лагранжа. Кроме того, наша теория дает возможность сформулировать
вторую теорему о среднем в интегральном исчислении, так как она является следствием диф-
ференциальной теоремы.
Ключевые слова: методика, теорема, среднее значение, функция, интеграл, производная,
теорема Лагранжа.
У статті сформульовано нове представлення відомої теореми у диференційному численні про середнє –
теореми Лагранжа. Прирощення функції представлено через похідні у кінцевих точках відрізку.
Середнє значення похідної по Лагранжу і нашої теорії співпадають, але не співпадають середні точки.
Наша теорія дозволяє знайти середню точку, що важко зробити на підставі теореми Лагранжа. Крім
того, наша теорема дає можливість сформулювати теорему про середнє у інтегральному численні, бо
вона просто є наслідком диференціальної теореми.
Ключові слова методика, теорема, середнє значення, функція, інтеграл, похідна,
теорема Лагранжа.
Introduction
Lagrange’s mean value theorem establishes a relationship between the increment of the
function on the interval and its derivative at some midpoint of a given segment. This relationship
is given by the formula (2) [1-3]. An important feature of this theorem is an analogue in integral
Mironenko L.P., Petrenko I.V.
«Искусственный интеллект» 2014 № 2 130
5M
calculus, which is called the first mean value theorem. In the integral calculus there is a second
mean value theorem, but there is no such analogue in the differential calculus. This "injustice"
can be eliminated, when Lagrange’s theorem is formulated in terms of derivatives at the segment
endpoints.
1 The Formulation and the Proof
of the Main Mean Value Theorem
The Main Mean Value Theorem. Let a function )(xfy satisfies the following con-
ditions: 1) it is continuous on a segment ],[ ba , 2) it is differentiable on the interval ),( ba and has
a finite one-sided derivatives )0( af and )0( bf , 3) the derivative )(xf is monotonic on
the interval ),( ba . Then there exists a unique point ),( ba for which takes place the equality
))(())(()()( bbfaafafbf . (1)
Proof. Consider the simplest case, when the derivative does not change its sign on the
interval ),( ba (Fig. 1). Figure 1 shows, that by virtue of monotony of the derivative )(xf , the
inequality tgBtgtgA takes place. If we multiply it by the difference )( ab and take
into account )()()( afbfabtg , we will obtain cfafbfсg )()( . By virtue of a
continuity of the function )(xf the sum ehfk accepts a number of continuous values from
сg up to cf , and therefore there is such point , for which the equality )()( afbfehfk
takes place. As far as ))(()( aafatgAfk and ))(()( bbfbtgBeh
the formula (1) takes place. The third condition of this theorem guarantees uniqueness of
the point .
Figure 1 – A geometric method for proving the theorem, ,gAcА eBhB
The second geometrical way of the proof of the theorem is easier than the first one.
We shall divide both parts of equality (1) into the difference )( ab and we shall enter a
parameter
ab
at
, where :10 t .10 ),1)(()()()(
ttbftaf
ab
afbf
Taking into account the geometrical meaning of a derivative, we obtain the equality
)1( ttgBttgAtg . It is necessary to consider an inequality tgBtgtgA , which
is provided with monotony of the derivative )(xf . Existence of the point follows from
Weierstrass theorem [2] according to which continuous function on a segment has a
continuous number of values between )(af and )(bf .
A New Representation of Lagrange’s Theorem in Differential Calculus
«Штучний інтелект» 2014 № 2 131
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2 The Formulation of Lagrange’s Theorem
and Compare it With the Main Mean Value Theorem
Lagrange's Theorem. Let a function )(xfy satisfies the following conditions: 1) it is
continuous on a segment ],[ ba , 2) it is differentiable on the interval ),( ba . Then there is a point
),( ba , for which there is the following equality
))(()()( abfafbf . (2)
When comparing equations (1) and (2) we obtain
))(())(())(( abfbbfaaf . (3)
Note that these two theorems midpoints are different, i.e. . Let's find a
function for which the points and will coincide. If in the formula (3) to take ,
you will get a simple differential equation
))(())(())(( abfbbfaaf
After integrating it with the variable we obtain
.)()(
2
)()()(
2
С
ab
bbfaaf
ab
bfaff
This is parabola, whose derivative is a linear function. From our theorem easily there
is the point :
)()(
)()()()(
bfaf
bbfaafafbf
. (4)
Note that by Lagrange's theorem it to make much harder.
From here the important application of our theorem follows. In the discrete
mathematics nodes, values of functions and its derivatives are set discretely. By Lagrange's
theorem discrete function needs to be approximated by a smooth function, as a rule, a
polynomial [4]. According to our theorem midpoints are calculated by the formula (4) and
it does not run any questionable approximations in the form of polynomials.
Example: 534 2 xxy , )3,1(x .
Find the point by Lagrange’s theorem: 2)1( y , 40)3( y , 38 xy ,
38)1()3( yy , 19
13
)1()3(
yy , 38)( y , 21938 .
By our theorem it is calculated by formula (4): 11)1( y , 27)3( y .
.2
2711
8111240
)3()1(
3)3()1()1()3(
ff
ffff
This example demonstrates the coincidence of midpoints for the parabola by Lagrange and
by our theory.
3 Roll’s And Cauchy’s Theorems in the New Edition
Roll’s Theorem. Let a function )(xfy satisfies the following conditions: 1) it is
continuous on a segment ],[ ba , 2) it is differentiable on the interval ),( ba , 3) the derivative
)(xf is monotonic on the interval ),( ba , 4) )()( afbf . Then there exists a unique point
),( ba for which takes place the equality
0))(())(( bbfaaf . (5)
Mironenko L.P., Petrenko I.V.
«Искусственный интеллект» 2014 № 2 132
5M
If to take into account )()( afbf , then the formula (5) follows from formula (1).
The expression for the point
)()(
)()(
bfaf
bbfaaf
is followed from the formula (5). The same
expression can be obtained from the formula (4), provided that )()( afbf .
Cauchy’s Theorem. Let functions )(xfy and )(xg satisfy the following condi-
tions: 1) they are continuous on a segment ],[ ba , 2) they are differentiable on the interval ),( ba ,
3) the derivatives )(xf and 0)( xg are monotonic on the interval ),( ba . Then there exists a
unique point ),( ba for which takes place the equality
))(())((
))(())((
)()(
)()(
bbgaag
bbfaaf
agbg
afbf
. (6)
Proof. In the proof we use the method of undetermined Lagrange’s multiplier [1]. Let
us introduce an auxiliary function )()()( xgxfxF and find the uncertain factor to
run the condition of Roll’s theorem: )()( aFbF . We will get
)()(
)()()()()()(
agbg
afbfagafbgbf
. (7)
According to Roll's theorem there is a point , for which takes place
0))(())(( bbFaaF
or
0)()()()()()( bbgbfaagaf .
Taking into account the expression (7), after simple transformations we obtain the
formula (6).
4 Simple Application of our Theory
is the Second Integral Mean Value Theorem
If we apply the formula (1) to the case when the function )(xf acts as its antiderivative,
i.e. )()( xfxF , then we immediately get the integral analogue of our theorem
))(())(()( bbfaafdxxf
b
a
.
This is so-called the second mean value theorem of integral calculus. Its generic
equivalent is obtained by presenting this formula in the form
dxbfdxafdxxf
b
a
b
a
)()()(
and replacing the integral measure dxxgxdGdx )()( [5]. Thus, finally, we obtain
dxxgbfdxxgafdxxgxf
b
a
b
a
)()()()()()(
.
Conclusions
The formula (2) is not the unique form of representation of Lagrange's theorem. The
variant of representation of an increment of a function through its derivatives at the endpoints of
a segment in the form of the formula (1) is possible still. Such formulation of Lagrange’s
theorem does not replace it, but even more expanding its capabilities. For example, the article
provides an example of application of the formula (1) in the integral calculus - it is a relatively
simple formulation of the second mean value theorem.
A New Representation of Lagrange’s Theorem in Differential Calculus
«Штучний інтелект» 2014 № 2 133
5M
References
1. Kudryavtsev L.D. Matematicheski anakiz / Kudryavtsev L.D. – Nauka, 1970. – Tom I. – 571 s.
2. Герасимчук В.С. Курс классической математики в примерах и задачах / В.С. Герасимчук,
Г.С. Васильченко, В.И. Кравцов. – Донецк, 2002. – 528 с.
3. Фихтенгольц Г.М. Курс дифференциального и интегрального исчисления / Фихтенгольц Г.М. –
М.: Наука, Изд-во ФМЛ, 1972. – Том 2. - 795 с.
4. Чисельні методи комп’ютерного аналізу / [Мироненко Л.П., Воропаєва В.Я., Локтіонов І.К.,
Турупалов В.В.]. – ДВНЗ «ДонНТУ», 2012. – С. 213.
5. Интегральные теоремы о среднем. Подход, основанный на свойствах интегральной меры /
Л.П. Мироненко, И.В. Петренко, О.А. Рубцова // Искусственный интеллект. – 2010. – № 4. – С. 617-622.
RESUME
L.P. Mironenko, I.V. Petrenko
A New Representation of Lagrange’s Theorem
in Differential Calculus
Background: in the integral calculus there is a second mean value theorem, but there is no
such analogue in the differential calculus. This "injustice" can be eliminated, when
Lagrange’s theorem is formulated in terms of derivatives at the segments endpoints.
Materials and methods: we used methods of differential calculus, derivative geometric
meaning, the notion of arithmetic mean value and weighted mean value, uncertain
Lagrange’s multipliers.
Results: the main result of the paper is the wording of the new mean value theorem in
differential calculus, which in turn gives rise to a number of theorems analogues of Rolle
and Cauchy. Сlassical Rolle's, Lagrange’s and Cauchy’s theorems are based on the concept
of a usual mean and arithmetic mean values. The new theorems are based on the concept of
the weighted mean value.
Conclusion: it is found a new representation of the mean value Lagrange’s theorem in the
differential calculus. The change of any functions is expressed in the terms of the
derivatives at the end points of a given closed interval. Mean values of the function are
coincided for both theories, but the midpoints are different. Our theory allows find the
midpoint easily. Such possibility is absent in Lagrange’s theorem. Besides our theory is
fundamental for the second mean value integral theorem.
The paper was received by editiorial 05.04.2014
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